Project 4 Fibonacci Sequence

Purpose: To investigate sequences. Outline: You will use cards to determine how many rabbits you have at any particular time. You will then explore this sequence of numbers. Content Objectives: Sequences, Fibonacci numbers. Materials Needed: Rabbit cards, calculator. Instructions: Go through the following exercises, and answer the given questions completely. You must show all work.

1. Consider the following scenario: a. We were given rabbits that were just born… one male and one female. b. Rabbits can mate at the age of one month, so by the end of the second month, each female can produce another pair of rabbits. c. We will assume (when ready) each female will produce one female and one male per month. d. The rabbits never die.

2. Use the ‘rabbit cards’ on the last page to model the problem for each month, until you use up all your cards. To keep track, the babies are one color (green) and the 1-month old pairs are another color (gray).

To help you start, you are given baby rabbits (month 0)

Next month (month 1), your baby rabbits did not have any babies, but are ‘mature’

Next month (month 2), your female rabbits (you only have one) has one pair of babies, so you have the original pair, and one set of babies.

3. Continue in this way, and write down how many rabbit pairs you have each month in the chart below.

Month Number of adult pairs Number of baby pairs Total pairs 0 0 1 1 1 2 3 4 5 6 7

4. What pattern do you see in this sequence of numbers? Explain.

5. Using this pattern, write out the sequence in the table below. If you need to also fill in the calculation used. 1

This sequence is quite famous, and is called the Fibonacci Sequence.

Leonardo Fibonacci was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages." He is best known to the modern world for the spreading of the Hindu-Arabic numeral system in Europe, primarily through the publication in the early 13th century of his Book of Calculation, the Liber Abaci; and for a number sequence named after him known as the Fibonacci numbers, which he did not discover but used as an example in the Liber Abaci. He was born around 1170 to Guglielmo, a wealthy Italian merchant. Guglielmo directed a trading post (by some accounts he was the consultant for Pisa) in Bugia, a port east of Algiers in the Almohad dynasty's sultanate in North Africa (now Bejaia, Algeria). As a young boy, Leonardo traveled with him to help; it was there he learned about the Hindu-Arabic numeral system.[5] Recognizing that arithmetic with Hindu-Arabic numerals is simpler and more efficient than with Roman numerals, Fibonacci traveled throughout the Mediterranean world to study under the leading Arab mathematicians of the time. Leonardo returned from his travels around 1200. In 1202, at age 32, he published what he had learned in Liber Abaci (Book of Abacus or Book of Calculation), and thereby popularized Hindu-Arabic numerals in Europe. Taken from Wikipedia, 5/18/2011 at 7:40pm PST.

Let’s see where we might find this number in the world around us…

6. Flower petals: Count the number of petals on each of these flowers. What numbers do you get? Are these Fibonacci numbers?

7. Seed heads: Look closely at the illustration. Do you see how the seeds form spirals? Find a spiral going toward the right or the left. Count how many spirals there are in the picture. Are they Fibonacci numbers?

8. Fill in the values again from #5 (the Fibonacci numbers) in the left column. Next, divide the number below by the number above, and write the result in the right column.

1 1 1/1 = 1

9. What do you notice as you go further down the column? Have we seen this value before?